Will I get a satisfying answer to the probability/philosophy question in the description?
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Under the Bayesian interpretation of probability, why is it be unfair for someone who isn't a player to secretly change the top card of the deck in a poker tournament? No player has gained any additional information about the deck, so it's still random to them.

Clearly a game of poker would still be fair if all the spectator did was look at the deck. So why does making a change (that none of the players can predict) become a problem?

The best answer I have at the moment is that if the change is known to the other spectators, it's changed the game for them because now the probabilities they assign to each player winning have changed away from a traditional game of poker.

But this implies that as long as the spectator makes the change secretly and no one else knows it happened, there's nothing unethical about that.

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I contend that the answer depends on the process the spectator uses to change the card. The players all share certain beliefs about the shuffling process used to prepare the deck; the spectator's card swap is unfair if it violates the players' foundational shared belief about the deck preparation process.

Below are two beliefs about the shuffling process; depending on which one you (or rather the players) think is the important one, the spectator has more or less leeway in the way they change the top card.

Belief 1: the shuffling process does not favor any individual player in the way it permutes the cards in the deck[*]. If the spectator changes the top card with another one in a way that enables them to influence the outcome of the game in favor a player of their choice, they have violated this belief. (This allows the spectator to use some processes that involve game state, like "swap the top card with the highest card not yet dealt", which would lead to surprising results if repeated many times, but would not favor any particular player.)

Belief 2: the shuffling process used to prepare the deck for the game results in a permutation of cards selected uniformly at random from all permutations (or close enough to equal that it's impossible to take advantage of the difference)[*]. If the spectator changes the top card in any way other than at random, they have violated this belief. ("any way other than at random" is poor phrasing, since a process like "swap the top two cards" does not violate this belief, but hopefully what I mean is clear enough, and I don't have the time to refine this just now)

[*] If you reject these as relying on the existence of "true" probabilities, here are Bayesian reframings:
1) the players have a very strong prior about the distribution of "who holds the strongest hand" they would observe over very many deals
2) the players have a very strong prior about the distribution of deck permutations they would observe over very many shuffles

and please try to apply similar reframings if you think I've snuck "true" probabilities in anywhere else.

predicted YES

:(

@IsaacKing did you consider this?

@jcb It doesn't really address my question. If the players don't find out about the swap, their credences have not changed.

@IsaacKing Is this not tantamount to "If a tree falls in the forest and no one to observes it, does it make a sound?"

predicted YES

@IsaacKing It's true that the players' credences have not changed, but the players do not care only about credences, they care about the reality of what the cards are and how those cards were selected.

If you're about to play russian roulette, and unbeknownst to you have I have loaded additional bullets into the revolver, I haven't done anything to change your credence about what happens when you pull the trigger, but I surely have made the game unfair to you!

@jcb Ooooh, yeah, that's a much better example of the problem.

it's changed the game for them because now the probabilities they assign to each player winning have changed away from a traditional game of poker.

This is true of anyone that has knowledge of the top of the deck including the spectator acting in secret and regardless of whether they change the top of the deck. If anyone has knowledge of the top of the deck then there is a possibility of that information leaking, and from an information perspective the only way to ensure there is no info leak is to ensure no one has the information to leak in the first place.

How might the secretive spectator leak the information?

• Intentionally, because they are collaborating with a player in the game.

• Body language, emotional response

• Side bets or meta-markets on the outcome of the game.

How can you prove that information has not been leaked to a hyper-observant meta spectator?

I would posit that it is unfair to even look at the deck order that a game would be played with. Changing the top card is only neutral if nobody knows the order anyhow, including thr spectator, but the concern is confidence and integrity, why would anyone want to change the top of the deck unless they had information or sought to gain information and it is the actual or precived information disparity which is problematic to calling the game fair.

So yeah, changing the top of the deck is going to be called unfair.

@ShitakiIntaki I think you're missing the point of the question.

@IsaacKing I guess so. What was the point? (I apologize if I am still off point, I only skimmed a bit of the preface of the lesswrong article linked. I may read more when I am off work.)

tldr; in a perfect idealized world it is not unethical to change the top card of the deck, in the real messy world it is unethical.

If the secretive spectator can get away with changing the top card of the deck, is it immoral? I suppose their action is not unethical, if they can guarantee that the information cannot be leaked and that no one knows it happened at all.

If a player could similarly change the top card of the deck with out anyone finding out about it, they could get away with cheating, but would it still be moral if they decided that they would, and actually could in fact, take all actions exactly as they would have if they did not have access to privileged information and the game is there for not impacted.

I suppose any one with godlike capacities could change the top card of the deck with out it being an immoral act, if we are allowed to suppose such a clinically sterile setting for our thought experiments.

Because the deck has five aces now.

predicted YES

I think the nature of this market requires that you respond to any unsatisfactory answers. I'd appreciate if you edited the question to include a maximum response time

@ArunChebrolu It resolves NO if I don't have such an answer by the first time I check the comments after the end date. I make no guarantees as to my response time, but I plan to check it a few times most days.

Does the person who made the change know the players, have any preference for who should win, or have any way of communicating with them?

@Nosaix Not that the players are aware of.

@IsaacKing Ahh, then it seems quite straightforward, at least from my point of view. Let's see if my argument is convincing enough for you.

If I am a player in this game, when I am deciding on my next move, I need to assess probabilities. I'm no poker player, but it's my understanding that it is possible to calculate a probability for whether one of the other players is likely to have a better hand than my own. This probability then informs my decisions on whether to stay in the game, raise my bets, etc.

But that probability is based on the assumption that the order of the cards is perfectly random and unknown, we can call this our prior probability.

When I see somebody come along and switch the top card, I need to consider whether this event has changed the probability.

If I knew that person was acting without any kind of intention to alter the course of the game, then there would be an argument for treating it as an essentially random change, with no impact on the probability. But, if there is even a chance that they are intentionally trying to influence the outcome of the game, then I do need to alter my prior probability to take this into account.

It doesn't matter whether the person switching cards actually does intend to change the game. Based on the player's incomplete knowledge, there is a non-zero probability that something has been done to deliberately alter the game, so there must be an impact, a new posterior probability that takes the new information into account. And this change from the prior to the posterior probability will then have an influence on how I act and therefore will change the course of the game.

The ethics side is a bit more tricky, because it does depend somewhat on your interpretation of ethics. But we have shown that a non-player has taken an action which has inescapably affected the course of the game. There has been an effect, regardless of their intentions, because players will alter their gameplay to account for the action. And furthermore, in the absence of evidence to the contrary, it would be entirely reasonable for one or more players to believe that cheating has occurred. And again, this is regardless of the actual intentions of the card switcher.

I believe, from most ethical perspectives it is straightforward that, for a non-player to alter the course of the game outside of the rues of that game is unethical. And to do so in a way that players may perceive as cheating only adds to that.

I appreciate I've only given a broad overview, so happy to discuss further if you're not convinced.

@Nosaix It only changes the probability the players calculate if they believe they have information about the spectator's motivations, which I've stated is not the case.

(My intended scenario in fact is that the players are unaware of the swap at all.)

The spectator who changes the top card can now insider trade with others on the outcomes of the game. This is unethical, unlike other types of insider trading, because the others cannot reasonably account for you knowing the top card. Because of this, I would argue that even looking but not changing the top card is unethical.

@ArunChebrolu What if the spectator places no bets?

predicted YES

@IsaacKing If the spectator places no bets, then it is ethical. However, remember that bets include more than explicit monetary transactions. Any conversation in which this spectator speculates on the outcome of the game is now giving them greater expected value, in terms of credibility as a predictor of game outcomes if not money.

@ArunChebrolu So if the spectator really wants Alice to win and changes the top card so that's guaranteed, you think that's fine?

predicted YES

@IsaacKing This is still an implicit bet on the outcome of the game. If Alice is a friend of the spectator's, they receive value from the tournament from Alice winning, and no (or negative) value from Bob winning.

@ArunChebrolu Hmm. So the spectator is unfairly taking away the enjoyment of those rooting for Bob to win. Interesting.

Why is this any less fair than Bob losing through normal means?

predicted YES

@IsaacKing When Bob loses normally, everyone has some true probability of who will win, because they assume a random deck. When you change the top card, you sway the probability in a way people cannot account for.

For example, if I were to tell people I would flip a standard coin, then flip a weighted coin, I have unethically changed the probability, even if I personally haven't bet on outcomes.

@ArunChebrolu That sounds like you're using the frequentist interpretation of probability, which isn't what I'm talking about here. Under Bayesianism there is no "true" probability; the state of the universe just is what it is and it evolves deterministically from that state to the next.

predicted YES

@IsaacKing You're right. By changing the top card you're not actually doing anything that violates Bayesian probability, which is impossible as far as I know. What you're doing here is just unethical, or frowned upon. When I say true probability I refer to the probability that others can reasonably infer given the premises.

@ArunChebrolu That's my question; why is it unethical if nobody's probabilities have changed? They haven't been negatively affected in any way!

Course, the same is true for a more straightforward example like stealing money from someone who never realizes they were stolen from.

predicted YES

@IsaacKing exactly right. At this point, we move away from math and into subjective morals. If you were in a coin flipping tournament and the fair coin was replaced by a weighted coin, would you consider it unfair?

@ArunChebrolu Unsure; seems like the same question as the deck of cards to me.

predicted YES

What if the replacement is done secretly, with only a 52/48 coin. If no one ever learns if it, is it unethical? I would say yes, because in the counterfactual where I'm betting, I would prefer the coin stated to be fair to be actually fair.

@ArunChebrolu Yeah, I'm just having trouble conceptualizing what that means. What if instead of replacing the coin, the spectator is simply mistaken about it being 52/48? Is that still unfair to the players? You're still treating the coin as though it has a single "true" probability, but that's not really the case. If the spectator places a different credence on the coin coming up heads than the players do, why is that a problem?

predicted YES

@IsaacKing If you think you are doing something unethical but are actually not, is it still unethical? Let's say you find a \$20 bill on the ground and pocket it. If it happens to be that bill spontaneously came into existence without hurting anyone (and created a corresponding amount of value in order to not hurt the economy), was it still unethical to take the bill? I would say the action was not ethical.